We study the position of the p-power Frobenius endomorphism \(\pi _E\) of a supersingular elliptic curve \(E/\mathbb {F}_{p^2}\) inside its quaternionic endomorphism algebra \({{\,\textrm{End}\,}}^0(E)\simeq B_{p,\infty }\) . Using the Weil polynomial \(T^2 - tT + p^2\) and Deuring’s description of \({{\,\textrm{End}\,}}^0(E)\) , we show that the Frobenius order \(R_E = \mathbb {Z}[\pi _E]\) is an imaginary quadratic order of discriminant \(t^2 - 4p^2\) whenever \(t^2\ne 4p^2\) , and we characterise its embedding into a maximal order. In particular, outside the degenerate cases \(t=\pm 2p\) , the minimal polynomial of \(\pi _E\) in \({{\,\textrm{End}\,}}^0(E)\) coincides with the Weil polynomial and no non-trivial linear relation holds in \(B_{p,\infty }\) . We briefly discuss how this description of \(R_E\) interacts with the structure of supersingular \(\ell \) -isogeny graphs and with arithmetic questions arising in isogeny-based constructions.
